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(String) Physics from (Higher) Algebra, II

Posted by Urs Schreiber

1) First of all, the phrasing in the previous entry suggested that the 2-category of bimodules internal to BiMod(R2 ) is equivalent to the 2-category of left R2 -modules when Ostrik’s theorem applies. Actually, what is pretty obvious is only that the former sits inside the latter in a certain sense. That’s good enough for the general argument I was interested in. Still, I suspect that the two really are equivalent, but that’s not so obvious. I have tried to cleanly describe the situation and the corresponding conjecture in these notes:

If anyone feels like helping to prove or disprove the conjecture stated at the very end of these notes, please let me know.

2) Second, the formulations in the previous entry didn’t make it clear which (2-)ring action the morphism in the various (2-)categories were supposed to respect. I give a new, somewhat more accurate re-formulation of the general idea below. Further refinements are obviously still necessary.

Finally, I should mention that Jeffrey Morton has a new preprint concerned with categorifying quantum mechanics:

Superficially this looks very different from what I am talking about here. But I think it might be related. However, I have not fully digested that paper yet.

So here is the first revision of the idea outlined last time:

1) 1-QM

Start by formulating ordinary quantum mechanics (QM) in an nicely algebraic way. There is the worldline of a particle, described by some geometric category P1 whose morphisms are 1-dimensional ‘spaces’ of some sort and whose composition is just gluing of such spaces.

To each object in P1 we want to associate a Hilbert space H of states. Being a vector space, let me regard H as a (left, say) ℂ-module.

But H has more structure. We also want the algebra of (bounded) observables being represented on it by bounded operators. We think of these as being some ring R of (ℂ-valued) functions over configuration space. H is hence supposed to be a (right, say) R-module, too.

So over each object of P1 we want a ℂ-R-bimodule. (Of course this is just a right R-module, but for the following it helps to think of it this way.)

Now, to a morphism in P1 we want to associate a linear operator, hence a morphism of left ℂ-modules. So we can say quantum mechanics is a functor

(1)QM:P1 →ℂ−R−lmod

where the target category is supposed to be that of ℂ-R bimodules with morphisms respecting only the left ℂ action.

This is straightforward to categorify and doing so yields all kinds of known structures that are being used elsewhere or have been proposed elsewhere.

2) 2-QM

Let P2 be some geometric 2-category modelling some sort of surface elements and horizontal as well as vertical gluing of them.

The target 2-category should have

- objects being C2 -R2 bimodules

where both C2 and R2 are “2-rings” but where R2 is like a 2-ring of 2-functions with values in C2 .

That’s easy. A 2-ring should be an abelian monoidal category. Let’s fix any such category and call it C2 . If C2 is sufficiently nice (see this entry on bimodules), we can apply Ostrik’s theorem and find that a (left) C2 -module is nothing but a category CA of right A-modules internal to C2 , where A is some algebra internal to C2 .

That’s cool, because we also have the category ACA of A-A-bimodules internal to C2 . This beast is also a 2-ring and it acts from the right on CA. Hence CA is actually a

C2 - ACA-(2-)bimodule

- morphisms in the target 2-category should be functors CA→CA which respect the left action by C2 on CA. By Ostrik’s theorem, this are nothing but (where ‘nothing but’ means up to the above mentioned conjecture) A-A-bimodules internal to C2

- 2-morphisms are natural trafos between these, which are internal bimodule homomorphisms (again, up to the above conjecture, see the above mentioned notes for details).

If we call this target 2-category T we have that 2-QM is a 2-functor

(2)2 −QM:P2 →T.

This is nice, for several reasons.

First, this concept actually appears (up to technicalities) in the work by Stolz and Teichner as a proposal for a formulation of 2D CFT. It is in principle known how it relates to path-integral prescriptions of field theory.

Second, I claim to have been able to prove large parts of my conjecture that “locally trivializing” such a 2-functor yields the structure of CFT formalism as it appears in the work by Fuchs-Runkel-Schweigert. See here for the latest version of this work.

Third, this does incorporate the approach proposed in HDAII, I think. This requires a little explanation:

One very nice categorical way to get internal modules is to use quivers. Let Q be any category and A(Q) its category algebra. An A(Q)-module in C2 is nothing but a functor

(3)Q→C2 .

An A(Q)-A(Q′) bimodule is a functor

(4)QxQ′op→C2 .

We can identify C2 with the category C(1 ) of functors from the category (1 ) containing a single morphism to C2 . Hence we may regard CA as the category of functors

(5)(1 )xQop→C2 .

But if we regard Q as a categorified version of a spacetime, this are nothing but the 2-functions considered in HDAII.

A module morphism is just a natural transformation between the functors Q→C2 .
The module tensor product over A is the functor AntiHom(−.−) which sends functors

Q→C2
and
Qop→C2

to the space of antinatural transformations between them. This also serves as the categorified inner product on CA, and hence really gives CA the structure of a 2-Hilbert space.

What is kind of remarkable is how the bimodule structure as well as the inner product actually become more natural (more functorial) after categorification.

Posted at January 24, 2006 7:44 PM UTC

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Re: (String) Physics from (Higher) Algebra, II

Re: (String) Physics from (Higher) Algebra, II

I’d be interested to hear how you think your ideas are related to those in Morton’s paper.

Yes, I would be interested in that, too!

I have spent two evenings over a beer reading Jeffrey Morton’s paper (after work, so to say) but I have made it only about half way through so far. I think I have absorbed everything before and including the beginning of section 5 “Stuff Types and Quantum Mechanics”.

I have browsed through the rest to get the basic idea, but will have to look at all that much more closely.

Given this incomplete understanding, I still dare to say the following:

Jeffrey Morton clearly deals with categorifying the Fock space formulation of QM. What I was envisioning, as sketched in the above entry, should, hopefully, incorporate this.

Let me maybe re-emphasize the simple idea:

QM associates Hilbert spaces to points and morphism between them to cobordisms between points.

2QM should associate 2-Hilbert spaces to points, morphisms of them to edges cobounding two points and 2-morphisms between these to surfaces cobounding these edges.

If we are being cavalier and do not presuppose too much extra structure, we can more concisely say that

- QM is a 1-functor from a geometric 1-category to the 1-category ℂ−Mod

- 2QM should be a 2-functor from a geometric 2-category to a 2-category C−Mod, where C is an abelian monoidal category.

My main point was that C−Modincludes the 2-category BiMod(C) and that hence this conception of 2QM does indeed have the general structure that appears in functorial formulations of 2-dimensional field theory.

But, of course, there can probably be 2-Hilbert spaces (or closely related structures) that are not, or not manifestly, equivalent to some module category over C.

As far as I understand, in Jeffrey’s paper a certain kind of 2-Hilbert spaces is constructed that can be thought of as categorified Fock spaces.

I note that morphisms of such 2-Hilbert spaces are defined in terms of spans. In what I have sketched these would be bimodules. But I think I have seen that bimodules can be expressed in terms of spans? Maybe I am hallucinating, does anyone know?

It is certainly due to the fact that I have not fully read the paper yet, but does Jeffrey Morton define a 2-category of 2-Fock-spaces? In the appendix I see the 2-category of stuff types discussed. But then there is also a 2-category of stuff operators, which I had expected should form (instead) the morphism categories in the 2-category of 2-Fock spaces. Hm, I guess I should just sit down and read this more carefully…

Re: (String) Physics from (Higher) Algebra, II

Urs writes:

It is certainly due to the fact that I have not fully read the paper yet, but does Jeffrey Morton define a 2-category of 2-Fock-spaces? In the appendix I see the 2-category of stuff types discussed. But then there is also a 2-category of stuff operators, which I had expected should form (instead) the morphism categories in the 2-category of 2-Fock spaces.

Jeff defines a single categorified Fock space, which you should think of as a categorified version of the Hilbert space for a quantum harmonic oscillator with one degree of freedom. The objects in this “Fock space” are
called stuff types - they’re groupoids equipped with a functor to the groupoid of finite sets.

And, he considers operators from this Fock space to itself. These are called stuff operators - they’re groupoids equipped with two functors to the groupoid of finite sets.

One can easily generalize this to consider Fock spaces for harmonic oscillators with n of freedom, using the groupoid of “n-tuples of finite sets”. But Jeff doesn’t mention this except near the very end of his paper.

The point that’s confusing you is that even this one single categorified Fock space is already a 2-category! Its objects are stuff types, but there are morphisms between these and 2-morphisms as well… which have nothing to do with the stuff operators mentioned above.

Category theorists would call this kind of thing “slice 2-category” or (I prefer) “over category”. Whenever we have a category and a fixed object X we get a category of “objects over X”, i.e. objects equipped with a morphism to X. The morphisms are commuting triangles. Similarly, whenever we have a 2-category and a fixed object X, we get a 2-category of objects over X.

Taking the 2-category to be Gpd and X to be the groupoid of finite sets, we get the 2-category of stuff types… which is our categorified Fock space for a harmonic oscillator with one degree of freedom.

Re: (String) Physics from (Higher) Algebra, II

The point that’s confusing you is that even this one single categorified Fock space is already a 2-category

Right, this comes a little bit as a superpise to me. I would expect a single categorified Fock space to be a 1-category, namely a 2-Hilbert space (or somehing very similar at least) as in your HDAII.

In fact, if I replaced the natural transformation alpha in the commuting triangles in Jeffrey’s paper (like (77)) everywhere with the identity transformation, would any of the main conclusions be affected? Would not the operators A and A* still act the way they should?

Re: (String) Physics from (Higher) Algebra, II

So as John mentions, the 2-category of stuff types is the categorification of Fock space developed in that paper. I agree that it’s sort of surprising that it is a 2-category rather than a regular category. This is a result of constructing it as an over category in Gpd, itself already a 2-category. Since we need groupoids (which are 1-categories) to have non-integer cardinalities, this seems to be an essential feature. Since we’re only interested in it as a 2-Hilbert space, I never really say much in there about the extra level of morphisms it has. It would be interesting to know what significance the extra level has, but offhand I don’t.

It would be fairly simple to develop a 2-category (really a 3-category, I suppose) of these categorified Hilbert spaces. The 2-category of stuff operators will then be the morphism category of endomorphisms of StuffTypes. The way I would define this is suggested in the conclusions and suggestions at the end of the paper. Every object in Gpd has an over category, um, over it just as FinSet0 does - in that special case, I cal it StuffTypes, but for a general groupoid G the over category is just “Groupoids over G” (Gpd/G). This consists of functors into G from other groupoids. So for each groupoid one gets a 2-Hilbert space in just the same way (though the physical interpretation will be different in each case). Then between each pair of these over categories, say associated to G1 and G2, there will be a 2-category of spans between G1 and G2, taking elements of the one 2-Hilbert space to the other by the same weak pullback construction as with stuff operators.

So I think it’s much as you expected, except that there is that extra level of morphisms, which had to be there to make cardinalities (and in particular, transition amplitudes) work out correctly (all those natural transformations alpha ensure that the inner product has the right decategorification). Again, I wish I could say more about this fact - maybe there is a more easily understood reason why the expected 2-categories are actually 3-categories.

One or Two

I see why we would want to have groupoids over sets as a way to categorify an ordinary function, and why that leads to a 2-category of such categorified functions, instead of merely a 1-category.

While I can follow this reasoning, it makes me wonder. It might be something about using spans instead of ‘dual’ formulations.

Let me try to clarify what I mean. I’ll adopt the simple example suggested by John in another comment.

Suppose I wanted to describe categorified QM in terms of Kapranov-Voevodsky 2-vector spaces.

In that case, I would set

(1)C=Vect

(where Vect is supposed to be the category of finite dimensional vector spaces over some fixed field K) and consider the 2-category

(2)CMod

of C-modules. In the present example this would be our 2-category of 2-Hilbert spaces.

There are at least two ways to describe this 2-category. One in terms of spans, the other, ‘dual’ to that, in terms of quiver representations.

Using the first, the 2-Hilbert spaces (the objects in CMod) would be categories of ‘half spans’ (over-categories) of the form

(3)E→n,

where E is a vector space and n is the category with n objects and only identity morphisms on these.

This is nothing but a vector space associated to each integer 1 ≤i≤n, which is the same as a KV-2-vector which is the same as a vector bundle over Obj(n).

In terms of spans, the 1-morphisms of CMod (‘KV-matrices’) are true spans

(4)M→n↓n

and 2-morphisms in CMod are morphisms of these (1-)spans.

I’ll get to the point of why I am recalling all this in just a moment. Before that, I note the description of CMod dual to the one above.

Instead of looking at half-spans E→n we can consider functors

(5)n→Vect.

This is a special case of realizing an algebra module in terms of a representation of its associated quiver. Here the quiver is the discrete one n, with no nontrivial edges on it.

From this point of view the objects of CMod are functor categories Vectn. 1-morphisms in CMod now are realized in terms of functors

(6)n×nop→Vect.

These act on the 2-vectors n→Vect by using AntiHom(−,−) for contractions.

Finally, 2-morphisms are natural transformations of these functors.

OK, so this are two dual and equivalent descriptions of QM categorified using KV 2-vector spaces. The reason why I am recalling all this at length is the following:

A straightforward generalization of the above scenario is obtained by replacing n by a more interesting category, Q, say, in particular by one which has non-trivial morphisms. A true quiver (from the second point of view).

But this has interesting consequences:

In terms of the second picture, that using quiver reps Q→Vect, nothing changes as far as the general formalism is concerned. We still have 1-categories of functors Q→Vect playing the role of categorified Hilbert spaces and so on.

On the other hand, in terms of spans everything now jumps by one dimension. We can no longer regard the half-span

(7)E→Q

as a vector bundle. Instead, this now is some sort of 2-bundle! (Strictly speaking, it was also a 2-bundle before, but a degenerate one which we could regard as a mere 1-bundle).

Consequently, we now seem to want weak over categories, weak pullbacks and so on, as you describe in your paper.

So, while I understand all the technical steps, I feel like I am missing some important general point here. Somehow the categorification dimensions don’t match.

Maybe we should ignore the topmost morphisms in the 2-span formulation? Maybe we should instead regard ordinary 1-Hilbert spaces already as span 1-categories?? I am confused.